deformation and fracture of rocks

8/2/2019 Deformation and Fracture of Rocks

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Deformation and Fracture of Rocks

2007 Taylor & Francis Group, London, UK


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CHAPTER 1

Precise measurements of fracture strength of rocks underuniform compressive stress

The conventional compression test, in which a short, right cylinder is loaded axially,is one of the most widespread experimental procedures in rock mechanics. This con-figuration is used in studies of both brittle and ductile behavior, in long term creep

studies as well as in studies of elastic behavior. In view of this wide usage, it is rathersurprising to find so little concern with what are widely recognized as bad featuresof this test (Seldenrath and Gramberg, 1958; Fairhurst, 1961).

In the typical arrangement, for example, the steel of the testing machine contactsthe rock cylinder, as shown in Fig. 1.1. Because steel and rock have different elastic

properties, radial shearing forces are generated at the interface when load is appliedto the rock sample. For most rocks these act inward and produce a clamping effect atthe end of the cylinder (Filon, 1902). This causes two things to happen. First, because

of the abrupt way in which the shearing stress changes near the outer edge of thesteel-rock interface, a stress concentration arises (Nadai, 1924). Second, if a fracturepropagates into the region near the end of the sample, growth of the fracture may be

Figure 1.1. Various methods of testing rock samples under uniaxial compression conditions.

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4 Experimental rock mechanics

impeded. These two effects might influence the apparent strength of a typical rockdifferently: the stress concentration would tend to lower, while the clamping wouldtend to raise the apparent strength. Almost certainly, the effects do not cancel eachother out.

Onecanthinkofanumberofwaystoeliminatetheseeffectsandtherebyimprovethecompression test. As a simple method to eliminate the clamping end effect caused byfriction on the steel-rock interface, various kinds of lubricants were applied betweenthe rock end surface and the steel platen, as shown in Fig. 1.1 (2). In this case, anumber of vertical cracks developed starting from the end surface of rock sample.This phenomenon seems to occur because of the intrusion of soft lubricator into therock specimen. The compressive strength obtained by this method is different fordifferent lubricants. Therefore, this method is not recommended.

Another simple method might be to make contact with the cylinder of rock througha metal which had identical mechanical properties. Then, no shearing forces wouldexist at the testing machine-rock interface, and, hence, there would be no stressconcentration at the corners. However, faults might still be checked at the interface,for the typical short specimens that are used. The length/diameter ratio is usuallyaround 2 so that any fault with a natural inclination of less that about 25 to themaximum compression would intercept one of the end faces. Actually, it is difficultif not impossible to match mechanical properties of rock with metals because mostrocks exhibit complex non-elastic features prior to fracture. Therefore, this approachholds little promise.

Another approach is the improvement of sample shape, whereby the effect of amismatch at the ends of the sample does not extend into the region of the samplewhere fracture occurs. One such design, suggested by Brace (1964), consists of adog-bone-shaped specimen with a reduced central section and large radius as shownin Fig. 1.1 (3). According to three-dimensional photoelastic analysis (Hoek, personalcommunication to Brace), the stress concentration in the fillets of these specimenswas extremely small and could be ignored. Of course, the stress concentration at the

steel-rock interface and the associated clamping effect still existed in this specimen,but these had no effect on what happened in the central section, which, in Bracesdesign, had an area one fourth of that of the ends. Braces specimens were quiteslender so that bending stresses were present. Rather than go to extreme measuresto eliminate them, they were simply evaluated by strain gages and the axial stressescorrected for bending in each experiments.

1.1 PRESENT SPECIMEN DESIGN

Braces specimens have one drawback in that they require a grinding technique moreelaborate than that used for producing straight cylinders. A new design was proposed

by Mogi (1966) which combines the better features of Braces specimen with greaterease of manufacture. The specimen (Fig. 1.2) is a long right cylinder connected to steel

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Precise measurements of fracture strength of rocks 5

Figure 1.2. The recommended test specimen design.

endpiecesbyepoxywhichisacommonlyavailablecommercialvarietywhichcontainsa filler of fine steel particles. The thickness of the epoxy is gradually decreased from

the steel end pieces toward the middle of the specimen to form a smooth fillet.The gradual decrease of the thickness of epoxy probably eliminates most of the

stress concentration at the contact of rock with steel. As the steel-filled epoxy has amodulus somewhat less than most rocks, the exact surface shape of the fillet is notcritical. The absence of a stress concentration near the end of the specimen is shown

by the way in which samples fractured. Fractures remained near the central sectionand did not enter the fillet region of the sample. No partially formed fractures could

be found in the fillet region. Yoshikawa and Mogi (1990) showed by a numericalcalculation that there is no stress concentration near the fillet region and the stressdistribution is homogeneous in the main part, as shown in Fig. 7.23.

Although the stress concentration near the ends is removed, the end part may give aclamping effect. To avoid the influence of clamping, the specimen has a length whichallows propagation of fractures completely within a uniform stress field. This criticallength is discussed in more detail below.



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With the longer sample required to avoid the clamping effect, bending becomeslikely, due mainly to a mismatch of loading and specimen axes. Bending was keptto a minimum here by (a) keeping the ends of the sample accurately parallel, (b) byapplying the load at a small central area of the upper steel piece (called loading axisadjuster in Fig. 1.2), and (c) by keeping the length of the end pieces as small as

possible. This is also discussed below.In summary, this sample design appears to be more practical than Braces specimen,

and it eliminates most of the bad features present in the conventional short cylinder.Yet, it is almost as easy to fabricate as the conventional specimen, as the rock part isstill a circular cylinder. Application of the method to extension tests will be mentionedin Section 3.2.b.

1.2 EFFECT OF LENGTH/DIAMETER RATIO ON APPARENTSTRENGTH AND FRACTURE ANGLE

As mentioned above, apparent strength in short specimens becomes higher due tothe clamping effect. With the increase of the length/diameter ratio, this effect shoulddecrease gradually and disappear at some critical value. Above this critical value,the strength should remain constant and should represent the true strength under uni-form compression.To obtain this value and the critical length/diameter ratio, relations

for uniaxial and triaxial compression. In these experiments, the triaxial testing appa-ratus designed by Brace was used for uniaxial and triaxial compression tests. Strainrate was held constant at about 104 sec1.

and fracture angles of Dunham dolomite, Westerly granite and Mizuho trachyteas functions of the length/diameter ratio using the above-mentioned test specimens(Mogi, 1966). Dunham dolomite and Westerly granite are compact and uniform in

structure, while Mizuho trachyte is rather porous (porosity 8.5%) with a non-uniformdistribution of pores. The results are summarized in Table 1.1 and Figs 1.3, 1.4 and

respectively.Dunham dolomite (Fig. 1.3). The average values of two or three strength mea-

surements and the standard deviation are indicated in the figure. The reproducibilityof strength (1 per cent or better) is very good. This is probably due to the homogeneousstructure of this rock and the high accuracy of the present measurement. The apparentstrength decreases markedly with the increase of L/D, but the value becomes nearlyconstant at high values of L/D. The critical value, (L/D)c, above which the apparentstrength becomes nearly constant is about 2.5. In this rock, the fracture is of the typ-ical shear type. The angle between the fracture and the loading axis also decreasesmarkedly with increase of L/D and becomes nearly constant (20) above (L/D)c. InFig. 1.3 (lower), large closed circles indicate angles of fault planes which go through


The author carried out careful measurements of the apparent compressive strength

between apparent strength and length/diameter ratio were experimentally investigated

1.5. Here, L and D are length and diameter of the central cylindrical part of specimen,
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Table 1.1. Apparent strength of specimens with different length/diameter ratio underuniaxial compression.

Rock L/D No. of Apparent strength, Relative strength,Specimens MPa %

Dunham dolomite 1.25 2 232 1 111.51.50 3 225 3 107.51.75 2 224 3 1072.00 3 219 3 104.52.25 2 214 0 102.52.50 2 209 0 1003.00 2 208 0 99.54.00 1 207 99

Westerly granite 1.25 2 263 9 109.5

1.50 2 252 2 1051.75 2 248 2 103.52.00 3 247 5 102.52.25 3 242 4 100.52.50 4 240 6 1003.00 2 239 4 99.53.50 2 238 6 994.00 3 238 5 99

Mizuho trachyte 1.00 1 126 115.51.50 1 114 104.5

1.75 1 112 1032.00 1 110 1012.25 1 112 102.52.50 1 110 1003.00 1 109 99.5

the whole sample and small closed circles indicate partial fault planes. In additionto actual faults, regular patterns of microfractures with no shear displacement were

observed in the central part of certain specimens. The angle between these microfrac-tures and the loading axis is indicated by open circles. Clearly, the orientation ofthe main faults depends on L/D, whereas the orientation of microfractures does not.Also, the former coincides with the latter for large L/D values. The dotted curve (1)in Fig. 1.3 (lower) gives a calculated value of from

cot = L/D (1.1)

where is an apparent fracture angle. Curve (2) is calculated from

cot = (L+ 0.25)/D (1.2)

and takes into account the actual intersection that a straight fault would havesome 3 mm above the base of the fillet, rather than exactly at the base of the f illetas in (1).



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Figure 1.3. Relations of apparent compressive strength (top figure) and apparent fractureangle (bottom figure) to length/diameter ratio in Dunham dolomite. In the bottom figure,large closed circle: good fault; small closed circle: partial fault; open circle: microfractures;dotted lines 1 and 2: calculated curves.

Figure 1.4. Relations of apparent uniaxial compressive strength (top figure) and apparentfracture angle (bottom figure) to length/diameter ratio in Westerly granite.

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Figure 1.5. Relation of apparent uniaxial compressive strength (top figure) and apparentfracture angle (bottom figure) to length/diameter ratio in Mizuho trachyte. Open circle:accompanied by bending.

The observed points are seen to lie somewhat between curves (1) and (2), whichsuggests that the fault angle in short specimens is strongly effected by clamping atthe ends of the specimens. That is, the fault is forced to run between opposite cornersof the sample.

Westerly granite (Fig. 1.4). The effect of L/D on strength and fracture angle isvery similar to that in dolomite, except for a greater fluctuation of strength values.This fluctuation (less than 3%) may be due to the larger grain size and high brittlenessof this rock. The apparent strength becomes constant in longer specimens. The critical

value of L/D is also 2.5. Apparent fracture angles also change with L/D and becomeconstant (19) in longer specimens. In this case, the typical fracture surface was notso flat as in the dolomite and no microfractures were observed.

Mizuho trachyte (Fig. 1.5). The apparent strength also decreases with the increaseof L/D and becomes constant above a critical value (2.0) of L/D, but this value issmaller than those obtained for the above-mentioned dolomite and granite. The finalvalue of the fracture angle is larger (25) than the other two. In this rock, a bend-ing fracture was sometimes produced at higher values of L/D; such a fracture forms

perpendicular to the specimen axis. The apparent strength of the other two was con-siderably lower than the normal case, as indicated by open circles in Fig. 1.5 (upper).The bending may have been caused by an uneven stress distribution in the specimendue to the porous structure.

Thus, the effect of L/D on apparent strength and fracture angle for these threerocks is important in shorter specimens. Above a critical value of L/D, strength and



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Figure1.6. Relation betweenfracture angle and the critical length/diameter ratio above whichthe end fix effect disappears. Broken lines 1 and 2: calculated curves; D0: dolomite (confining

pressure= 0.1 MPa); G0: granite (0.1 MPa); G1: granite (17 MPa); T0: trachyte (0.1 MPa).

fracture angle were independent of L/D. This final value of strength and fracture angleprobably represents the strength and fracture angle under uniform compressive stressfor the following reasons: (1) The fracture starts and ends inside the central part ofspecimen. Therefore, stress concentrations at the end have no effects on strength.(2) Strength and fracture angle are independent of L/D, so the clamping effect is alsoavoided. (3) If bending is important, apparent strength should decrease with increaseof the length of specimen. Here, since such a strength decrease is small in longerspecimens (L/D< 4), so that bending effect is probably not significant.

The relation between the critical length/diameter ratio (L/D)c and the final fracture

angle 0 is presented in Fig. 1.6. Curves (1) and (2) in this figure are calculated asgiven above. The agreement between the observation and the calculation is good,especially for curve (2).

A number of researchers have investigated the effect of length/diameter ratio onapparent uniaxial compressive strength for rocks (e.g., Obert et al., 1946; Dreyer et al.,1961) and for concrete (e.g., Gonnerman, 1925; Johnson, 1943; A.C.I., 1914). These

previous experiments were carried out employing the conventional method in whichstraight circular cylinders or prisms were used, and therefore stress concentrations atthe end of specimen were probably present. In Fig. 1.7, some results of these pre-vious experiments (bottom figure) are presented together with the above-mentionedresults (top figure). The apparent strength considerably decreases with the increase oflength/diameter ratio (L/D). The strong effect of L/D in shorter specimens (L/D< 2)is clearly caused by the clamping effect. And these previous experiments did not givea constant final value of strength and fracture angle for large L/D. Apparent strength



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Figure 1.7. Comparison of the authors results (Mogi, 1966) (top figure) with some previous

experiments (bottom figure). The relative strength is recalculated for strength at L/D= 2.5.Obert et al. (1946): average for various rock types; Dreyer et al. (1961): rock salt; A.C.I.(1914), Gonnerman (1925), Johnson (1943): concrete.

still continues to decrease with L/D above the expected critical length/diameter ratio.This could have been due to bending caused by a mismatch of specimen and loadingaxis, or by the non-uniform structure of rock and concrete.

One example in the previous reports on similar measurements on an andesite by

ShimomuraandTakata(1961)ispresentedintheleftfigureofFig. 1.8. The fluctuationof strength values (50%) in their experiment is much greater than that of the above-mentioned Dunham dolomite shown in the right figure. From this experimental result,it is impossible to obtain any significant information. Such great fluctuation may bedue mainly to high inhomogeneity of the rock samples. These previous experimentalresults presented in Fig. 1.7 and Fig. 1.8 show that a suitable selection of mechanicallyuniform rock samples is very important in rock mechanics.

1.3 COMPARISON WITH THE CONVENTIONAL METHOD

As mentioned above, the conventional test, which uses short cylinders, is accompaniedby stress concentration and the clamping effect. Specimens of length/diameter ratio 2were used in most previous experiments. According to the present results, such tests



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Figure1.8. Simplified relations of apparent strength (relative values) to length/diameterratio.Left figure: andesite (Shimomura and Takata, 1961); right figure: Dunham dolomite (Mogi,1966).

may give considerably higher strength and larger fracture angle. Furthermore, sincethe end parts of the cylinder are laterally fixed to a hard end piece, the length of thecentral part where the stress is nearly uniform is still shorter than the actual length. For

comparison with the present method, some uniaxial compression tests were carriedout using the conventional method. In Dunham dolomite, the strength measured was231 MPa and the fracture angle was 30. These values are 10 per cent and 50 per centhigher, respectively, than those obtained using the present method. For Westerly gran-ite, strength and fracture angle were 250 MPa and 26. These are 5 per cent and 36 percent higher than those obtained using the present method. Thus, applying the conven-tional test to such hard rocks may give considerably higher strength and larger fractureangle. In Mizuho trachyte, for which the true fracture angle is higher, the difference

between the revised method and the conventional method was not appreciable.

Thus, the precise measurements of true compressive strength and fracture angleunder uniaxial compression can be best carried out by use of the revised specimendesign which is a longer cylinder with a smooth epoxy fillet at the end of rockspecimen, as shown in Fig. 1.1 (4). And it should be noted that the length/diameterratio should be larger than 2.5.

1.4 DECREASE OF THE END EFFECTS BY CONFINING PRESSURE

It was expected that end effects might change with confining pressure. To inves-tigate this, apparent strength and fracture angle of the specimens having variouslength/diameter ratio were measured under confining pressure (Mogi, 1966). Theshape of specimen and the experimental procedure are similar to the case of uniaxial



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Figure 1.9. Relation of apparent strength (top figure) and apparent fracture angle (bottomfigure) to length/diameter ratio in Dunham dolomite under confining pressure of 18 MPa (a)and 33 MPa (b).

compression, except for jacketing of the specimen to prevent the intrusion of pressedoil. Corrections were made for the frictional force between the axial piston and theo-ring seal of the pressure vessel. (The experiments under confining pressure arediscussed in detail in the following chapter.)

Dunham dolomite. The results are presented in Figs. 1.9 (a) and 1.9 (b). Under18 MPa confining pressure, the f inal fracture angle and the critical value of L/Dare nearly the same as in the uniaxial case. However, the effect of L/D on strengthis considerably smaller than in the uniaxial case. Strength and fracture angle under33 MPa confining pressure is nearly independent of L/D.

Westerly granite. The effect of L/D was determined at a confining pressure of17 MPa, 51 MPa, and 108 MPa. With the increase of pressure, the effect of L/D on

strength and fracture angle decreases, as shown in Figs. 1.10 (a)1.10 (c). The fractureangle gradually increases with confining pressure and the critical value of L/D alsodecreases.

Thus, it is concluded that end effects decrease gradually with an increase of pressureand become very small under confining pressure greater than 3050 MPa, as shownin Fig. 1.11. This decrease of end effect may be due to (1) the relative decrease ofthe effect of lateral restriction at the end part by increase of lateral pressure, (2) theincrease of fracture angle under pressure, and (3) the increase in the ductility of rocks.The observations show that the shape of the specimen becomes less critical at highpressure and so the conventional triaxial compression test may also give as nearlycorrect results as the method used here.

This discovery of the marked decrease of the end effect by confining pressureprovided a key for the design of the true triaxial compression machine (Mogi, 1970),which is explained in Chapter 3.

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Figure 1.10. Relation of apparent strength (top figure) and apparent fracture angle (bottomfigure) to length/diameter ratio in Westerly granite under confining pressure of 17 MPa (a),51 MPa (b) and 108 MPa (c).

Figure 1.11. Relation of apparent strength to length/diameter ratio in Dunham dolomite (topfigure) and Westerly granite (bottom figure).

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REFERENCES

Am. Concrete Inst. (A.C.I.) (1914). Report of committee on specifications and methods oftests for concrete materials. Proc. Am. Concrete Inst., v. 10, p.422.

Brace, W. F. (1964). Brittle fracture of rocks. In: State of stress in the Earths Crust. Judd, W.R.(ed.) New York: Elsevier, 111174.

Dreyer, W. and H. Borchert. (1961). Zur Druckfestigkeit von Salzgesteinen. Kali undSteinsalz, Heft 7, S.234241.

Fairhurst, C. (1961). Laboratory measurement of some physical properties of rock. In 4thSymp. on Rock Mechanics, Bull. Mineral Industries Expt. Sta. Penn. State Univ., No. 76.Hartman, H. L. (ed.), 105118.

Filon, L. N. G. (1902). On the elastic equilibrium of circular cylinders under certain practicalsystems of load. Phil. Trans. R. Soc., London, Ser. A 198, 147233.

Gonnerman, H. F. (1925). Effect of size and shape of test specimen on compressive strength

of concrete. Proc. A. S. T. M., v. 25 (Part 2), p.237.Johnson, J. W. (1943). Effect of height of test specimen on compressive strength of concrete.

A. S. T. M. Bulletin, No. 120, p.19.Mogi, K. (1966). Some precise measurements of fracture strength of rocks under uniform

compressive stress. Felsmechanik und Ingenieurgeologie 4, 4155.Nadai, A. (1924). ber die Gleitund Verzweigungsflchen einiger Gleichgewichtszustnde,

Z. Physik 30, p.106.Obert, L., S.L. Windes andW. I. Duvall. (1946). Standardizedtests for determining thephysical

properties of mines rocks. U. S. Bur. Mines. Rep. Invest., no. 3891, p.1.Seldenrath, Th. R. and J. Gramberg. (1958). Stress-strain relations and breakage of rocks.

In: Mechanical Properties of Non-Metallic Materials. Walton, W. H. (ed.). London:Butterworths, 79102.

Shimomura, Y. and A. Takata. (1961). On the mechanical behaviors and the breaking mech-anism of rocks (1st Report) Shape and scale effect on fracture of rock specimensin compression. J. Mining and Metallurgical Inst. of Japan, Vol. 77, No. 876, 914.(in Japanese)

Yoshikawa, S. and K. Mogi. (1990). Experimental studies on the effect of stress history ofacoustic emission activity A possibility for estimation of rock stress. J. Acoustic Emission,8, 113123.

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